As far as I know, Anim8or simply displays all codecs available on your computer. You can install and un-install most codecs just like any other software...
The way codecs work should not be much related to Anim8or - some codecs produce reasonable quality and file size, other codecs are simply useless, but it all depends on the settings and codec versions... Also keep in mind that the same codec that was used in production must be available to play the video. Therefore it's only reasonable to use the most common codecs. Like Kreator said, you should try XVid and DivX, these two codecs are quite common and produce a reasonable image quality.
However the best you can do IMO is to avoid all .avi codecs completely. I usually export my videos uncompressed or as a series of .bmp images to get the best possible quality before the post-production starts. When I have my final video done, I decide which format/codec to use...
There are 3 most common file formats for videos - AVI, MPEG and WMV. Most video-editing applications support all of them plus many more (only .avi is supported by Anim8or though). As written above, the .avi format causes problems with codec compatibility. MPEG is a bit better from this point of view, but I also had some problems with .mpg videos... My favorite format is .wmv, mainly because the file size/quality ratio is IMO the best. Although it's a proprietary format developed by Microsoft, it is possible to play .wmv videos on most platforms (using the right program) and I've never had problems with compatibility, codecs, versions etc... WMV videos can also be easily optimized for internet streaming. The only disadvantage is that you need to use a third-party video editing software since WMV is not supported by Anim8or directly. But like you said, even Movie Maker can do the conversion so this shouldn't be a problem.